Minnesota, which has won seven of nine, has not lost to Colorado in four games this season, including two shootout wins. Backstrom has three of those wins.
"It's seems to be a one-goal game every time we play against them," said Backstrom, who notched his 18th NHL shutout.
Peter Budaj had 21 saves for Colorado, which has lost three straight at home but only one in regulation.
The Wild scored the only goal they needed 5:57 in when Miettinen pounced on Andrew Brunette's rebound for his eighth goal.
"It kind of bounced right on my tape there," said Miettinen, who has seven goals and nine points in his last eight games.
Minnesota didn't think the early goal was going to stand up, but the Avalanche, coming off a 3-1-1 road trip, couldn't generate much offense. They had only 13 shots in the final 40 minutes and 23 overall despite having six power-play opportunities.
"We have to be better on the power play," Avalanche coach Joe Sacco said. "We'll continue to work on it and we'll get better at it. It was probably the difference in the game."
That and the play of Backstrom. He wasn't asked to do much, but when Colorado turned up the pressure in the third period he came up big.
Marek Svatos had two good chances in the first six minutes, and then had another chance on a breakaway with 7:20 left, but Backstrom made the saves.
"The offense is there but we can't rely on scoring four goals every night," Backstrom said. "Sometimes you have to find a way to win and battle through the defense."
The Avalanche had several chances to tie it in the first, but they couldn't capitalize on three power-play opportunities.
"We took a lot of penalties, we spent a lot of time in the box, but our penalty killers did a great job," Minnesota coach Todd Richards said "Your goalie has to be your best penalty killer, and he was very good tonight. Nik made some big, big saves for us when we needed him to."
Colorado thought it got even when Matt Duchene beat Backstrom with 14:37 left in the second, but Avalanche forward T.J. Galiardi and Minnesota winger Martin Havlat battled near the net. The officials ruled the net was off its moorings when the puck crossed the line. The call was confirmed by video review.
"It was going in, obviously, and it ended up going in," Galiardi said. "But I think the net was off before it went in. I don't know if they pushed it off or what happened. I can only speculate."
Sacco thought the Wild dislodged the net but said the goal probably would not have counted.
"The ref said our player kicked the puck in. It was a kicking motion, so It probably would have been nullified anyway," he said. "That wasn't the difference in the game."
- The Wild are 12-1-2 in their last 15 regular-season games against the Avalanche.
- Colorado G Craig Anderson missed his fourth straight game because of a neck injury sustained at Florida on Dec. 2.
- The Avalanche were 0 for 6 on the power play. It was the second time in nine games they failed to score a power-play goal.
- Avalanche C Justin Mercier played in his first NHL game.